FIG. 1 is a block diagram of a conventional signal receiving system 10. The signal receiving system 10 comprises a signal retriever 140, a demapper 160 and a decoder 180. The demapper 160 comprises a mapping function mapping apparatus 164 and a quantizer 167.
The signal retriever 140 receives an input signal and transforms a time-domain input signal to two corresponding signals namely a frequency-domain inphase signal (I signal) and quadrature signal (Q signal). The demapper 160 generates the digital data corresponding to I and Q signals according to constellations applied to the input signal. For example, the constellations applied to the modulation, such as binary phase shift keying (BPSK), 16 quadrature amplitude modulation (16QAM) and 64QAM, are different, so the I and Q signals corresponding to the digital data are different. Lastly, the decoder 180 transforms the digital data to an output data.
Theoretically, the I and Q signals generated by the signal receiving system 10 should map correctly on the constellations to two integers of a Gray code, which is a coding method and is a set of a sequence. Each number of the Gray code is represented by binary, and there is only one different bit between any two of Gray code. However, the signal processed by the signal receiving system 10 may be interfered by the noise such that the I and Q signal generated by the signal retriever 140 may not be an integer, i.e., the I and Q signal may not map exactly to the Gray code on the constellation, such that one needs other methods for mapping the non-integer I and Q signals to the Gray code.
One of the solutions to solve the above problem is a soft decision method. FIG. 2 is a conventional 64QAM constellation, wherein the I-axis represents the I signal, and the Q-axis represents the Q signal. Each point on the constellation maps to a 6-bit value (0 to 26−1), of which the first three bits represent the I part, and the last three bits represent the Q part. If the signal receiving system 10 uses the 64QAM, the soft decision method is to map the coordinates (I, Q) of the I and Q signals received by the demapper 160 to a soft coordinate (I0, I1, I2, Q0, Q1, Q2). For example, the I coordinate of 5.3 maps to (I0*, I1*, I2*)=(5.3, −1.3, 0.7) by the mapping function mapping apparatus 164. The corresponding mapping function is as follows:
  {                                                        I              0              *                        =            I                                                                          I              1              *                        =                                          -                                                    I                                                              +              4                                                                                      I              2              *                        =                                          -                                                                                                                    I                                                              -                    4                                                                                +              2                                               
Limited by the memory in the hardware, practically, one needs to quantize a decimal to a value acceptable to the hardware. Therefore, (I0*, I1*, I2*)=(5.3, −1.3, 0.7) is quantized to (I0, I1, I2)=(3, −2, 2) as shown in FIG. 2 by quantizer 167, where (I0, I1, I2)=(3, −2, 2) is very different to the original (I0*, I1*, I2*)=(5.3, −1.3, 0.7). The conventional soft decision method is to divide I0, I1 and I2 on the constellation into N equal parts (N=8 in FIG. 2) without taking the distinct ranges of I0, I1 and I2 into consideration. That is, when I0 is determined as being positive or negative, the total range of I1 is only a half of the total range of I0, i.e., the total range of I1 is only the positive region or the negative region of I0. Similarly, when I1 is determined as being positive or negative, the total range of I2 is only a half of the total range of I1. More specifically, respective absolute distances from I0=4 and I1=4 to the origin are not identical. In fact, from FIG. 2, the distance between origin and I1=4 is a half of the absolute distance between origin and I0=4. Hence, dividing all I0, I1 and I2 into N equal parts causes quantizing distortion to undesirably affect the determination of the decoder, such that not only the coding gain is reduced but also the bit error rate (BER) is increased from being unable to accurately correct erroneous bits. Therefore, it is urgently needed a better soft decision method and associated signal receiving system to increase the coding gain and to reduce the bit error rate.